Abstract
In this paper, we obtain several new characterizations of the Clifford torus as a Lagrangian self-shrinker. We first show that the Clifford torus \({\mathbb {S}}^1(1)\times {\mathbb {S}}^1(1)\) is the unique compact orientable Lagrangian self-shrinker in \({\mathbb {C}}^2\) with \(|A|^2\le 2\), which gives an affirmative answer to Castro–Lerma’s conjecture in Castro and Lerma (Int Math Res Not 6:1515–1527; 2014). We also prove that the Clifford torus is the unique compact orientable embedded Lagrangian self-shrinker with nonnegative or nonpositive Gauss curvature in \({\mathbb {C}}^2\).
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Acknowledgments
The authors would like to thank the referee for the very careful review and for providing a number of valuable comments and suggestions. The first author was supported in part by NSFC Grant No. 11271214. The second author was supported in part by NSFC (Grant Nos. 11201243 and 11571185) and “Specialized Research Fund for the Doctoral Program of Higher Education, Grant No. 20120031120026”.
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Li, H., Wang, X. New Characterizations of the Clifford Torus as a Lagrangian Self-Shrinker. J Geom Anal 27, 1393–1412 (2017). https://doi.org/10.1007/s12220-016-9723-x
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DOI: https://doi.org/10.1007/s12220-016-9723-x